Fixed point indices of iterates of orientation-reversing homeomorphisms

TL;DR

This paper demonstrates that any integer sequence satisfying Dold's congruences can be realized as the fixed point index sequence of an orientation-reversing homeomorphism in higher dimensions, providing a complete classification for boundary-preserving cases.

Contribution

It constructs explicit examples of orientation-reversing homeomorphisms with prescribed fixed point index sequences and classifies fixed point indices for boundary-preserving maps.

Findings

Any sequence satisfying Dold's congruences can be realized.

Complete description of fixed point indices for boundary-preserving maps.

Construction of homeomorphisms with prescribed fixed point index sequences.

Abstract

We show that any sequence of integers satisfying necessary Dold's congruences is realized as the sequence of fixed point indices of the iterates of an orientation-reversing homeomorphism of $\mathbb{R}^{m}$ for $m\geq 3$. As an element of the construction of the above homeomorphism, we consider the class of boundary-preserving homeomorphisms of $\mathbb{R}^{m}_{+}$ and give the answer to [Problem 10.2, Topol. Methods Nonlinear Anal. 50 (2017), 643 - 667] providing a complete description of the forms of fixed point indices for this class of maps.